1answer.
Ask question
Login Signup
Ask question
All categories
  • English
  • Mathematics
  • Social Studies
  • Business
  • History
  • Health
  • Geography
  • Biology
  • Physics
  • Chemistry
  • Computers and Technology
  • Arts
  • World Languages
  • Spanish
  • French
  • German
  • Advanced Placement (AP)
  • SAT
  • Medicine
  • Law
  • Engineering
nikklg [1K]
3 years ago
13

Determine whether the sequences converge.

Mathematics
1 answer:
Alik [6]3 years ago
4 0
a_n=\sqrt{\dfrac{(2n-1)!}{(2n+1)!}}

Notice that

\dfrac{(2n-1)!}{(2n+1)!}=\dfrac{(2n-1)!}{(2n+1)(2n)(2n-1)!}=\dfrac1{2n(2n+1)}

So as n\to\infty you have a_n\to0. Clearly a_n must converge.

The second sequence requires a bit more work.

\begin{cases}a_1=\sqrt2\\a_n=\sqrt{2a_{n-1}}&\text{for }n\ge2\end{cases}

The monotone convergence theorem will help here; if we can show that the sequence is monotonic and bounded, then a_n will converge.

Monotonicity is often easier to establish IMO. You can do so by induction. When n=2, you have

a_2=\sqrt{2a_1}=\sqrt{2\sqrt2}=2^{3/4}>2^{1/2}=a_1

Assume a_k\ge a_{k-1}, i.e. that a_k=\sqrt{2a_{k-1}}\ge a_{k-1}. Then for n=k+1, you have

a_{k+1}=\sqrt{2a_k}=\sqrt{2\sqrt{2a_{k-1}}\ge\sqrt{2a_{k-1}}=a_k

which suggests that for all n, you have a_n\ge a_{n-1}, so the sequence is increasing monotonically.

Next, based on the fact that both a_1=\sqrt2=2^{1/2} and a_2=2^{3/4}, a reasonable guess for an upper bound may be 2. Let's convince ourselves that this is the case first by example, then by proof.

We have

a_3=\sqrt{2\times2^{3/4}}=\sqrt{2^{7/4}}=2^{7/8}
a_4=\sqrt{2\times2^{7/8}}=\sqrt{2^{15/8}}=2^{15/16}

and so on. We're getting an inkling that the explicit closed form for the sequence may be a_n=2^{(2^n-1)/2^n}, but that's not what's asked for here. At any rate, it appears reasonable that the exponent will steadily approach 1. Let's prove this.

Clearly, a_1=2^{1/2}. Let's assume this is the case for n=k, i.e. that a_k. Now for n=k+1, we have

a_{k+1}=\sqrt{2a_k}

and so by induction, it follows that a_n for all n\ge1.

Therefore the second sequence must also converge (to 2).
You might be interested in
Following are the prices of 12 tickets listed on the Ticket Racket ticket broker site for a Taylor Swift concert.
Ludmilka [50]

Let me use Python to find the answers

Program.

\tt import\:statistics \:as\:stats

\tt Prices=[75,120,120,145,150,150,150,175,175,200,225,275]

\tt print(stats.mean(Prices))

\tt print(stats.median(Prices))

\tt print(stats.mode(Prices))

Output:;

  • Mean=163.333
  • Median=150.0
  • Mode=150

Console attached

8 0
3 years ago
Read 2 more answers
Tay-Sachs disease is a genetic disorder that is usually fatal in young children. If both parents are carriers of the disease, th
LuckyWell [14K]

Answer:

Step-by-step explanation:

Given that Tay-Sachs disease is a genetic disorder that is usually fatal in young children. If both parents are carriers of the disease, the probability that each of their offspring will develop the disease is approximately 0.25.

Let X be the no of children having this disease out of 4 occasions

Then X has two outcomes and each trial is independent of the other trial

Hence X is binomial with n =4, and p = 0.25

a) P(X=4) = 4C4 (0.25)^4 = 0.003906

b) P(X=1) = 4C1 (0.25)(0.75)^3\\= 0.4219

c) P(4th child/3 children did not) = 0.25 since independent.

4 0
3 years ago
8 1/3 minus 3 3/4 please help
Novosadov [1.4K]
4 7/12 is your answer :)
6 0
3 years ago
Read 2 more answers
What is an equivalent expression to 4(5a+2-3b)
marishachu [46]
20a+8-12b. Could be written as 20a-12b+8
5 0
2 years ago
Which graph represents the solution to the inequality y<1/2x-4
wlad13 [49]
<h3>Answer: Lower left corner</h3>

The idea is you graph the line y = (1/2)x-4, which is the same as y = 0.5x-4

This line goes through (0,-4) and (8,0) as the y and x intercepts. Make this line a solid line (due to the "or equal to" portion of the inequality sign). Then shade below this solid line to show the region of points that have smaller y coordinates compared to y = (1/2)x-4

Any point in the shaded region makes y \le \frac{1}{2}x-4 true

5 0
3 years ago
Other questions:
  • Lucero wants to hang 3 paintings in her room. The widths of the paintings are 10 1/2 inches, 3 1/2 feet, 2 feet, and 2 3/4 inche
    11·1 answer
  • Melinda earns 5 minutes to play video games for every hour she spends on homework.
    5·2 answers
  • * * <br> Urgent I really must get these I would appreciate any help I can get, thank you. (22 pts)
    13·1 answer
  • Does anybody know where i can watch Naruto shippuden dub online for free no downloads
    13·2 answers
  • Help me with this please!!!
    8·1 answer
  • A student has started a lawn care business. He charges $15 per hour to mow lawns and $20 per hour for gardening. Because he is s
    5·1 answer
  • Is (2, 3) a solution to this system of equations?<br> 2x + y = 7<br> 17x + 19y = 18<br> yes<br> no
    10·1 answer
  • How do you factor this problem? I know you have to pull out the GCF first but I'm still stumped. 18x³-51x²+36x​
    7·1 answer
  • Which of the following situations can be represented by the expression 3n – 15 ? Select all that apply. ​
    14·1 answer
  • What is 5x + 12 &lt; 2 ?
    15·2 answers
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!